Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior

Abstract

Consider a regular $d$-dimensional metric tree $\Gamma$ with root $o$. Define the Schroedinger operator $-\Delta -V$, where $V$ is a non-negative, symmetric potential, on $\Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-\gamma }$ at infinity, where $1<\gamma \le d\le 2,\gamma \ne 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-\Delta -V$. In other words, we will describe the asymptotical behavior of $\inf \sigma (-\Delta -\alpha V)$ as $\alpha \to 0+$

References in corpus (3)

• Kirchhoff's Rule for Quantum Wires
• Weakly coupled states on branching graphs
• Schrödinger operator on homogeneous metric trees: spectrum in gaps